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JOURNAL OF RESEARCH of the National Bureau of Standards— Mathematical Sciences Vol. 80B, No. 3, July-September 1976 One-Sided Tolerance Limits for the Normal Distribution, P= 0.80, 7 = 0.80 Roy H. Wampler Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234 (June 24, 1976) A table is given of factors k used in constructing one-sided tolerance limits for a normal distribution. This table was obtained by interpolation in an existing table of percentage points of the noncentral ^-distribution. The accuracy of the table is estimated, and a compar- ison is made of the presently computed factors with a previously published approximation. Key words: Noncentral ^-distribution; normal distribution; statistics; tolerance limits. 1. Introduction Let X be a normal random variable with mean ijl and standard deviation a. If /i and a are known, we can say that exactly a proportion P of the population is below /d-\-K p a, where K v is a normal deviate defined by -j=f*\xv(-t*/2)dt=P. (1) If /x and cr arc unknown, one can estimate these quantities from a random sample of n observations: x u x 2 , • • •, $n> The mean ix is estimated by _ 1 n % = Z 2—1 X ii n i=i and the standard deviation a is estimated by The problem, is now to find k such that the probability is y that at least a proportion P of the population is below x-\-ks. Mathematically, the problem is to find k such that Pr{Pr(X<x+ks) >P} =7 (2) where X has a normal distribution with mean ju and standard deviation <r, and P and 7 are specified probabilities. As is indicated in Owen [10], eq (2) can be written as Pr{T f <kji\i=K 9 ji}=y where T f denotes the noncentral /-distribution with f=n— 1 degrees of freedom and with noncentrality parameter 343 2. Existing Tables For many specified values of P and 7, tables of factors k for one-sided tolerance limits as denned in (2) above have been published. References to these tables are listed in Owen [10] and Johnson and Kotz [4], chapter 31. Since none of the tables cited in these references gives exact values of k in the case where P = 0.80 and 7=0.80, the present table has been prepared to fill this gap. Approximations to the factors k for this case were included in table III of Owen [8]. 3. Computing Method for the Present Table Exact values of k can be computed from the appropriate percentage points of the non- central ^-distribution. Table III (pp. 214-237) of Locks, Alexander, and Byars [7] gives 3-decimal percentage points of the noncentral t for: /=t&— 1 = 1(1)20, 25, 30, 35, 40; ^=0.00(0.25)3.00; €=1-7=0.01, 0.05(0.05)0.95, 0.99, 0.995. By interpolating on the values in this table for e = 0.20, one can obtain the factors k correspond- ing to P=0.80, 7 = 0.80. Table 1 presented here gives one-sided tolerance factors k, Owen's approximate values of k and the relative error in the approximate values, for n— 2(1)21, 26, 31, 36, 41. The values of k were obtained from the 3-decimal percentage points of the noncentral ^-distribution given by Locks et al. [7], using five-point Lagrangian interpolation. Four-point Lagrangian interpolation yields the same values of k (to 3 decimals) as does five-point interpolation. These computations were done through the use of OMNITAB (Hogben et al. [2]). For 71=4(1)12, the values of k were also computed using the first five terms of Stirling's interpolation formula as given on page 71 of Kunz [6]; again the same values of k (to 3 decimals) were obtained. For all these calculations the value of ^=^.80=0.84162123 was taken from a table of the inverse normal probability distribution (Kelley [5]). Table 1. One-sided tolerance limit factors for the normal distribution Values of k such that Pr {Pr (X<z+hs) >P} =7 for P=0.80, 7 = 0.80 n k Approx. k Rel. error 2 3.420 2. 37544 0.305 3 2.016 1. 70985 . 152 4 1. 675- 1. 50952 .099 5 1. 514 1. 40392 .073 6 1.417 1. 33609 .057 7 1.352 1. 28781 .047 8 1.304 1. 25119 . 040 9 1.266 1. 22219 .035 10 1.237 1. 19849 .031 11 1.212 1. 17866 .028 12 1. 192 1. 16175 .025 13 1. 174 1. 14711 .023 14 1. 159 1. 13427 . 021 15 1. 145+ 1. 12290 .019 16 1. 133 1. 11274 .018 17 1. 123 1. 10358 .017 18 1. 113 1. 09528 .016 19 1. 104 1. 08771 .015 20 1.096 1. 08076 . 014 21 1.089 1. 07436 .013 26 1.060 1. 04855 .011 31 1.039 1. 02968 . 009 36 1.023 1. 01512 .008 41 1.010 1. 00346 .007 344 4. Estimated Accuracy Locks et al. [7] reported that numerous checks were made of their tables against previously published tables, and "in no case where comparison was made ... is the disagreement more than one unit in the last decimal place." In order to assess the accuracy of the Fs given here in table 1 for P=0.80, 7=0.80, the following checks were made. Starting with the 3-decimal percentage points in table III of Locks for 7 = 0.75 and 0.90, values of k for all possible combinations of 7 = 0.75, 7 = 0.90, P=0.75, P=0.90, and 71=2(1)21(5)41 were computed by five-point Lagrangian interpolation as described in section 3 above. These values were then compared with the exact 3-decimal values published by Owen [9], pp. 52 and 58. This permitted the comparison of 96 values of k. Of the 92 values corresponding to ti>2, 88 were in full (3-decimal) agreement, and 4 differed by 0.001. For 7i=2, larger discrepancies were found. The values of k obtained here differed from Owen's exact values as indicated below: n P 7 Interpolation in Locks: (A) Owen's Exact k: (B) Difference: (A)-(B) 2 0. 90 0.90 10. 264 10. 253 0. 011 2 . 90 .75 3.994 3. 992 . 002 2 . 75 . 90 5. 859 5. 842 . 017 2 . 75 . 75 2. 227 2. 225 . 002 One may infer from these comparisons that the value of k for /> = 0.80, 7=0.80, n=2 given in table 1 is probably not in error by more than about 0.017 and is probably larger than the exact value. For n>2, any errors in the computed values of k are probably in the neighbor- hood of 0.001. 5. Comparison With an Approximation The approximate values of k given here in table 1 are taken from table III of Owen [8]. This approximation was derived by Jennett and Welch [3] and was further discussed in chapter 1 of Eisenhart, Hastay and Wallis [1]. The formula for this approximation is k = K P +^K p 2 -ab where a=l- K y > 2(n-l) --K P 2 - K y and K y is defined in the same manner as was K v in eq (1). The relative error in the approximate values of k shown in table 1 was computed from the formula (approx. k)—k k Rel. error = We note that in all cases covered by this table the approximations are smaller than the values of k computed by the method described in section 3. This is in agreement with Owen's state- ment [8] that the approximation will probably underestimate k for 7 < 0.95. 6. References [1] Eisenhart, Churchill, Hastay, Millard W., and Wallis, W. Allen, (eds.). Techniques of Statistical Analysis (McGraw-Hill Book Co., New York, 1947). [2] Hogben, David, Peavy, Sally T., and Varner, Ruth N., OMNITAB II User's Reference Manual, Nat. Bur. Stand. (U.S.), Tech. Note 552, 264 pages (Oct. 1971). 345 [3] Jennett, W. J., and Welch, B. L., The control of proportion defective as judged by a single quality char- acteristic varying on a continuous scale, Supplement to the Journal of the Royal Statistical Society 6, 80-88 (1939). [4] Johnson, Norman L., and Kotz, Samuel, Continuous Univariate Distributions — 2 (Houghton Mifflin Co., Boston, Mass., 1970). [5] Kelley, Truman Lee, The Kelley Statistical Tables (Harvard University Press, Cambridge, Mass., 1948). [6] Kunz, Kaiser S., Numerical Analysis (McGraw-Hill Book Co., New York, 1957). [7] Locks, M. O., Alexander, M. J., and Byars, B. J., New Tables of the Noncentral t Distribution, Report ARL 63-19, Aeronautical Research Laboratories, Wright-Patterson Air Force Base, Ohio, 1963. [8] Owen, Donald B., Tables of Factors for One-Sided Tolerance Limits for a Normal Distribution, Mono- graph No. SCR-13, Sandia Corporation, 1958. [9] Owen, D. B., Factors for One-Sided Tolerance Limits and for Variables Sampling Plans, Monograph No. SCR-607, Sandia Corporation, 1963. [10] Owen, D. B., A survey of properties and applications of the noncentral /-distribution, Technometrics 10, 445-478 (1968). (Paper 80B3-449) 346